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CHAPTER 4: CLUSTERING OF AXIS AND XMS SOURCES
and D H is the Hubble distance:
D H = c H 0
. (4.4.7)
If we consider two sources separated in the sky an angle θ and with transverse
comoving distances to the observer D M1 and D M2 , respectively, the comoving
separation of the second object measured from the first object is ([161]):
r =
√
d 2 D 2 M1 + D2 M2 − 2dD M1D M2 cosθ (4.4.8)
where ([167], [148])
d =
√
1 + Ω k
(
H0 D M1
c
⎡ √
) 2
+ D M1cosθ
⎣1 −
D M2
1 + Ω k
(
H0 D M2
c
) 2
⎤
⎦ . (4.4.9)
In a spatially flat Universe (Ω k = 0), d = 1 and equation 4.4.8 reduces to the
cosine rule for Euclidean space:
r =
√
D 2 M1 + D2 M2 − 2D M1D M2 cosθ. (4.4.10)
The procedure to generate the random sample is very similar to that we followed
for the angular clustering. We formed a pool with real sources with detection
likelihood higher than 15 in the band under study and a secure redshift
identification. For each field, we drew a source from the pool and generated
its new position at random but keeping its distance to the optical axis of the
telescope (see section 4.3.1) and checked whether its count rate was above the
field sensitivity map. In this case, the source is kept, along with its redshift,
and a new source is drawn from the pool until the number of extracted random
sources is equal to the number of real sources in the field. If the source count
rate is below the sensitivity map in the new position, the source is discarded
and the procedure is repeated. Following this method, the average redshift distribution
of the simulated samples resembles the real distributions with great
accuracy (see Fig. 4.9).
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